ENVE-GEOE 224 - Online Lecture 8 - Complete

.pdf

School

University of Waterloo *

*We aren’t endorsed by this school

Course

224

Subject

Civil Engineering

Date

Oct 30, 2023

Type

pdf

Pages

Uploaded by deraokonkwo

2021-03-25 1 ENVE/GEOE 224: Probability & Statistics Online Lecture 8 – Introduction to Hypothesis Testing Prof. Philip J. Schmidt Department of Civil & Environmental Engineering, University of Waterloo Winter 2021 Lecture Outline 1. Developing hypothesis testing concepts from an illustrative example 2. Errors in hypothesis testing – Type I & Type II error 3. Tests concerning a single mean Our weekly joke will come later… 2 of 25 ENVE/GEOE 224 (W2021) – P. Schmidt 1 2

2021-03-25 2 Learning Objectives • Understand that the improbability of a result given a particular (null) hypothesis suggests that this hypothesis may be untrue • Be able to define Type I error – the probability of falsely rejecting the null hypothesis (on the basis of improbable experimental results) when it is actually true • Be able to define Type II error – the probability of failing to reject the null hypothesis when a particular alternative hypothesis is true • Know the steps of the hypothesis testing procedure, including critical statistics and P-values, and how to complete a test on a single mean 3 of 25 ENVE/GEOE 224 (W2021) – P. Schmidt The Coffee Contest Controversy 4 of 25 ENVE/GEOE 224 (W2021) – P. Schmidt There are many problems in which, rather than estimate the value of a parameter, we must decide whether a statement concerning a parameter is true or false. We must TEST A HYPOTHESIS about the parameter. Let’s begin with an example… A chain of coffee shops runs a contest in which it is claimed that there is a one-in-six chance of winning a prize with each purchased cup of coffee. Over the course of the contest, Bill drinks 60 cups of coffee, but only wins four times instead of the 10 expected. Could this be the result of pure chance? 3 4

2021-03-25 3 The Coffee Contest Controversy 5 of 25 ENVE/GEOE 224 (W2021) – P. Schmidt Assuming that the result is due to pure chance and that the reported probability of winning is true, the contest is comprised of Bernoulli trials with 𝑝 = 0.16 ̇ 𝑋 = # of wins in 60 cups of coffee, and is binomial with 𝑛 = 60 , 𝑝 = 0.16 ̇ 𝐸 𝑋 = 𝑛𝑝 = 10 𝑃 𝑋 ≤ 4 ≈ 0.0202 The Coffee Contest Controversy 6 of 25 ENVE/GEOE 224 (W2021) – P. Schmidt Because 𝑃 𝑋 ≤ 4 ≈ 0.0202 is small the data might suggest that the claim of one-in-six chances of winning is untrue and that 𝑝 < 0.16 ̇ Such an assertion presumes that additional data (from Bill drinking more coffee or considering the results of other coffee-drinkers) would continue to show evidence that 𝑝 < 0.16 ̇ They must have lied about the chances of winning! …or maybe not! 5 6

2021-03-25 4 The Coffee Contest Controversy 7 of 25 ENVE/GEOE 224 (W2021) – P. Schmidt If 𝑝 = 0.16 ̇ is true, then 𝑃 𝑋 ≤ 4 ≈ 0.0202 and about one in 50 people would be at least as unlucky as Bill! If 𝑝 = 0.16 ̇ is true, then 𝑃 𝑋 = 0 ≈ 1.77 × 10 ିହ and one million coffee-drinkers who drink 60 cups each would be expected to include about 18 who never won at all and vowed to never drink coffee again! If 𝑝 = 0.16 ̇ is true, then by chance Bill might instead have won 17 times and used 𝑃 𝑋 ≥ 17 ≈ 0.0164 to argue 𝑝 > 0.16 ̇ ! Bill’s data provide some evidence to support a claim different from the one made by the coffee shop chain, but we can’t quite be sure… [unless we tally the results for EVERY cup of coffee in the contest] The Coffee Contest Controversy 8 of 25 ENVE/GEOE 224 (W2021) – P. Schmidt This exercise was an example of a hypothesis test! Formulate hypotheses (“null” and “alternative”): → 𝐻 ଴ : the observations are purely the result of chance 𝑝 = 0.16 ̇ Asserts a particular value ( 𝑝 = 𝑝 ଴ , 𝜇 = 𝜇 ଴ , σ = 𝜎 ଴ , 𝜇 ଵ − 𝜇 ଶ = 𝛿 ) → 𝐻 ஺ : the observations are the result of a real effect 𝑝 ≠ 0.16 ̇ Can be either one-tailed or two-tailed ( 𝑝 < 𝑝 ଴ , 𝑝 > 𝑝 ଴ , 𝑝 ≠ 𝑝 ଴ ) We can’t use the alternative hypothesis 𝑝 < 0.16 ̇ here because the direction was determined by the data! We would have to state such a hypothesis before collecting the data! 7 8

2021-03-25 5 The Coffee Contest Controversy 9 of 25 ENVE/GEOE 224 (W2021) – P. Schmidt Identify a test statistic: → A test statistic is a statistic with which to assess the evidence of the data against the null hypothesis (often 𝑍 , 𝑇 , 𝜒 ଶ , or 𝐹 ) In this example, random variable 𝑋 is a test statistic with which to weigh the evidence against the null hypothesis 𝐻 ଴ : 𝑝 = 0.16 ̇ → Critical values of test statistics ( 𝑍 ௖௥௜௧ , 𝑇 ௖௥௜௧ , 𝜒 ௖௥௜௧ ଶ , etc.) are often determined using a fixed significance level such as 𝛼 = 0.05 More on this later…this approach is not used in this example The Coffee Contest Controversy 10 of 25 ENVE/GEOE 224 (W2021) – P. Schmidt Calculate a P-value: → A P-value is the probability of getting a value of the test statistic at least as extreme as the one observed if the null hypothesis is true In this example, 𝑃 𝑋 ≤ 4 ≈ 0.0202 gives a P-value of 0.0202 for 𝐻 ஺ : 𝑝 < 0.16 ̇ (if the data had not informed the choice of tail to test) It is better to say that 𝑃 𝑋 ≤ 4 ≈ 0.0202 gives a P-value of 0.0404 for 𝐻 ஺ : 𝑝 ≠ 0.16 ̇ , reflecting the presence of similarly extreme data in the other tail that would also suggest 𝑝 ≠ 0.16 ̇ 9 10

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version